July  2017, 16(4): 1253-1264. doi: 10.3934/cpaa.2017061

Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space

1. 

chool of Mathematics and Systems Science, Beihang University (BUAA), Beijing, 100191, China

2. 

School of Mathematics and Computer Science, Jiangxi Science and Technology Normal University, Nanchang, 330038, China

3. 

Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA

* Corresponding author: Zhao Liu

Received  July 2016 Revised  January 2017 Published  April 2017

Fund Project: The first two authors were partly supported by the NNSF of China (No. 11371056), the first author was also supported by the NNSF of China (No. 11501021), the third author was partly supported by a US NSF grant and a Simons Fellowship from the Simons Foundation.

In this paper, we investigate the Hardy-Sobolev type integral systems (1) with Dirichlet boundary conditions in a half space $\mathbb{R}_+^n$. We use the method of moving planes in integral forms introduced by Chen, Li and Ou [12] to prove that each pair of integrable positive solutions $(u, v)$ of the above integral system is rotationally symmetric about $x_n$-axis in both subcritical and critical cases $\frac{n-t}{p+1}+\frac{n-s}{q+1}\geq n-2m$ (Theorem 1.1). We also derive the non-existence of nontrivial nonnegative solutions with finite energy in the subcritical case (Theorem 1.2). By slightly modifying our arguments for studying the integral system, we can prove by a similar but simpler way that the same conclusions also hold for a single integral equation of Hardy-Sobolev type in both critical and subcritical cases (Theorem 1.3).

Citation: Wei Dai, Zhao Liu, Guozhen Lu. Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1253-1264. doi: 10.3934/cpaa.2017061
References:
[1]

G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations in ${\mathbb{R}^N}$ and $\mathbb{R}_N^ + $ through the method of moving plane, Comm. Partial Differential Equations, 22 (1997), 1671-1690.  doi: 10.1080/03605309708821315.  Google Scholar

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L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems, Disc. Cont. Dyn. Sys., 33 (2013), 3937-3955.  doi: 10.3934/dcds.2013.33.3937.  Google Scholar

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L. Cao and Z. Dai, A Liouville-type theorem for an integral equations system on a half-space $\mathbb{R}_n^ + $, J. Math. Anal. Appl., 389 (2012), 1365-1373.  doi: 10.1016/j.jmaa.2012.01.015.  Google Scholar

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G. CaristiL. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67.  doi: 10.1007/s00032-008-0090-3.  Google Scholar

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W. Chen and Y. Fang, Higher order or fractional order Hardy-Sobolev type equations, Bulletin of the Institute of Mathematics Academia Sinica (New Series), 9 (2014), 317-349.   Google Scholar

[6]

L. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equation with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.  Google Scholar

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D. Cao and Y. Li, Results on positive solutions of elliptic equations with a critical HardySobolev operator, Methods Appl. Anal., 15 (2008), 81-96.  doi: 10.4310/MAA.2008.v15.n1.a8.  Google Scholar

[8]

L. Chen, Z. Liu and G. Lu, Symmetry and regularity of solutions to the weighted HardySobolev type system, Adv. Nonlinear Stud. , 16 (2016), no. 1, 1-13. doi: 10.1515/ans-2015-5005.  Google Scholar

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W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. and Dyn. Sys. , Vol. 4,2010.  Google Scholar

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W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[11]

W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 4 (2009), 1167-1184.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar

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W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar

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W. ChenC. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Patial Differential Equations, 30 (2005), 59-65.  doi: 10.1081/PDE-200044445.  Google Scholar

[14]

W. Dai, Z. Liu and G. Lu, Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space, Potential Analysis, (2016). doi: 10.1007/s11118-016-9594-6.  Google Scholar

[15]

Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867.  doi: 10.1016/j.aim.2012.01.018.  Google Scholar

[16]

Y. Fang and J. Zhang, Nonexistece of positive solution for an integral equation on a half-space $\mathbb{R}_N^ + $, Comm. Pure Appl. Anal., 2 (2013), 663-678.  doi: 10.3934/cpaa.2013.12.663.  Google Scholar

[17]

Y. Guo and J. Liu, Liouville-type theorems for polyharmonic equations in ${\mathbb{R}^N}$ and in $\mathbb{R}_N^ + $, Proc. Roy. Edinburgh Sect. A., 138 (2008), 339-359.  doi: 10.1017/S0308210506000394.  Google Scholar

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B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in ${\mathbb{R}^N}$, Mathematical Analysis and Applications, vol. 7a of the book series Advances in Mathematics, Academic Press, New York, 1981.  Google Scholar

[19]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Patial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196.  Google Scholar

[20]

C. Jin and C. Li, Symmetry of solution to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670.   Google Scholar

[21]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar

[22]

Z. Liu and W. Dai, A Liouville type theorem for poly-harmonic system with Dirichlet boundary conditions in a half space, Advanced Nonlinear Studies, 15 (2015), 117-134.  doi: 10.1515/ans-2015-0106.  Google Scholar

[23]

C. Li, A degree theory approach for the shooting method, preprint, arXiv: 1301.6232. Google Scholar

[24]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.  doi: 10.1007/s002220050023.  Google Scholar

[25]

Y. Lei and C. Li, Decay properties of the Hardy-Littlewood-Sobolev systems of the LaneEmden type, preprint, arXiv: 1302.5567. Google Scholar

[26]

Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems, Discrete Contin. Dyn. Syst., 36 (2016), 3277-3315.  doi: 10.3934/dcds.2016.36.3277.  Google Scholar

[27]

Y. LeiC. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations, Calc. Var. and PDEs, 45 (2012), 43-61.  doi: 10.1007/s00526-011-0450-7.  Google Scholar

[28]

D. LiP. Niu and R. Zhuo, Symmetry and non-existence of positive solutions for PDE system with Navier boundary conditions on a half space, Complex Variables and Elliptic Equations, 59 (2014), 1436-1450.  doi: 10.1080/17476933.2013.854346.  Google Scholar

[29]

G. LuP. Wang and J. Zhu, Liouville-type theorems and decay estimates for solutions to higher order elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 653-665.  doi: 10.1016/j.anihpc.2012.02.004.  Google Scholar

[30]

G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Calc. Var. and PDEs, 42 (2011), 563-577.  doi: 10.1007/s00526-011-0398-7.  Google Scholar

[31]

G. Lu and J. Zhu, Axial symmetry and regularity of solutions to an integral equations in a half-space, Pacific J. Math., 253 (2011), 455-473.  doi: 10.2140/pjm.2011.253.455.  Google Scholar

[32]

G. Lu and J. Zhu, Liouville-type theorems for fully nonlinear elliptic equations and systems in half spaces, Advanced Nonlinear Studies, 13 (2013), 979-1001.  doi: 10.1515/ans-2013-0413.  Google Scholar

[33]

G. Lu and J. Zhu, The maximum principles and symmetry results for viscosity solutions of fully nonlinear equations, J. Differ. Eqs, 258 (2015), 2054-2079.  doi: 10.1016/j.jde.2014.11.022.  Google Scholar

[34]

L. Ma and D. Chen, A Liouville type theorem for an intergral system, Comm. Pure Appl. Anal., 5 (2006), 855-859.  doi: 10.3934/cpaa.2006.5.855.  Google Scholar

[35]

W. Reichel and T. Weth, A prior bounds and a liouville theorem on a half-space for higherorder elliptic Dirichlet problems, Math. Z., 261 (2009), 805-827.  doi: 10.1007/s00209-008-0352-3.  Google Scholar

[36]

J. XuH. Wu and Z. Tan, Radial symmetry and asymptotic behaviors of positive solutions for certain nonlinear integral equations, J. Math. Anal. Appl., 427 (2015), 307-319.  doi: 10.1016/j.jmaa.2015.02.043.  Google Scholar

[37]

J. Villavert, Shooting with degree theory: Analysis of some weighted poly-harmonic systems, J. Differ. Eqs., 257 (2014), 1148-1167.  doi: 10.1016/j.jde.2014.05.003.  Google Scholar

[38]

R. ZhuoF. Li and B. Lv, Liouville type theorems for Schrodinger system with Navier boundary conditions in a half space, Comm. Pure Appl. Anal., 3 (2014), 977-990.  doi: 10.3934/cpaa.2014.13.977.  Google Scholar

[39]

Z. Zhang, Nonexistence of positive solutions of an integral system with weights, Adv. Difference Equ. , 61 (2011), 10 pp. doi: 10.1186/1687-1847-2011-61.  Google Scholar

[40]

Y. Zhao, Regularity and symmetry for solutions to a system of weighted integral equations, J. Math. Anal. Appl., 391 (2012), 209-222.  doi: 10.1016/j.jmaa.2012.02.016.  Google Scholar

show all references

References:
[1]

G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations in ${\mathbb{R}^N}$ and $\mathbb{R}_N^ + $ through the method of moving plane, Comm. Partial Differential Equations, 22 (1997), 1671-1690.  doi: 10.1080/03605309708821315.  Google Scholar

[2]

L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems, Disc. Cont. Dyn. Sys., 33 (2013), 3937-3955.  doi: 10.3934/dcds.2013.33.3937.  Google Scholar

[3]

L. Cao and Z. Dai, A Liouville-type theorem for an integral equations system on a half-space $\mathbb{R}_n^ + $, J. Math. Anal. Appl., 389 (2012), 1365-1373.  doi: 10.1016/j.jmaa.2012.01.015.  Google Scholar

[4]

G. CaristiL. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math., 76 (2008), 27-67.  doi: 10.1007/s00032-008-0090-3.  Google Scholar

[5]

W. Chen and Y. Fang, Higher order or fractional order Hardy-Sobolev type equations, Bulletin of the Institute of Mathematics Academia Sinica (New Series), 9 (2014), 317-349.   Google Scholar

[6]

L. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equation with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.  Google Scholar

[7]

D. Cao and Y. Li, Results on positive solutions of elliptic equations with a critical HardySobolev operator, Methods Appl. Anal., 15 (2008), 81-96.  doi: 10.4310/MAA.2008.v15.n1.a8.  Google Scholar

[8]

L. Chen, Z. Liu and G. Lu, Symmetry and regularity of solutions to the weighted HardySobolev type system, Adv. Nonlinear Stud. , 16 (2016), no. 1, 1-13. doi: 10.1515/ans-2015-5005.  Google Scholar

[9]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS Book Series on Diff. Equa. and Dyn. Sys. , Vol. 4,2010.  Google Scholar

[10]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[11]

W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 4 (2009), 1167-1184.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar

[12]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar

[13]

W. ChenC. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Patial Differential Equations, 30 (2005), 59-65.  doi: 10.1081/PDE-200044445.  Google Scholar

[14]

W. Dai, Z. Liu and G. Lu, Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space, Potential Analysis, (2016). doi: 10.1007/s11118-016-9594-6.  Google Scholar

[15]

Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867.  doi: 10.1016/j.aim.2012.01.018.  Google Scholar

[16]

Y. Fang and J. Zhang, Nonexistece of positive solution for an integral equation on a half-space $\mathbb{R}_N^ + $, Comm. Pure Appl. Anal., 2 (2013), 663-678.  doi: 10.3934/cpaa.2013.12.663.  Google Scholar

[17]

Y. Guo and J. Liu, Liouville-type theorems for polyharmonic equations in ${\mathbb{R}^N}$ and in $\mathbb{R}_N^ + $, Proc. Roy. Edinburgh Sect. A., 138 (2008), 339-359.  doi: 10.1017/S0308210506000394.  Google Scholar

[18]

B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in ${\mathbb{R}^N}$, Mathematical Analysis and Applications, vol. 7a of the book series Advances in Mathematics, Academic Press, New York, 1981.  Google Scholar

[19]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Patial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196.  Google Scholar

[20]

C. Jin and C. Li, Symmetry of solution to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670.   Google Scholar

[21]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar

[22]

Z. Liu and W. Dai, A Liouville type theorem for poly-harmonic system with Dirichlet boundary conditions in a half space, Advanced Nonlinear Studies, 15 (2015), 117-134.  doi: 10.1515/ans-2015-0106.  Google Scholar

[23]

C. Li, A degree theory approach for the shooting method, preprint, arXiv: 1301.6232. Google Scholar

[24]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.  doi: 10.1007/s002220050023.  Google Scholar

[25]

Y. Lei and C. Li, Decay properties of the Hardy-Littlewood-Sobolev systems of the LaneEmden type, preprint, arXiv: 1302.5567. Google Scholar

[26]

Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems, Discrete Contin. Dyn. Syst., 36 (2016), 3277-3315.  doi: 10.3934/dcds.2016.36.3277.  Google Scholar

[27]

Y. LeiC. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations, Calc. Var. and PDEs, 45 (2012), 43-61.  doi: 10.1007/s00526-011-0450-7.  Google Scholar

[28]

D. LiP. Niu and R. Zhuo, Symmetry and non-existence of positive solutions for PDE system with Navier boundary conditions on a half space, Complex Variables and Elliptic Equations, 59 (2014), 1436-1450.  doi: 10.1080/17476933.2013.854346.  Google Scholar

[29]

G. LuP. Wang and J. Zhu, Liouville-type theorems and decay estimates for solutions to higher order elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 653-665.  doi: 10.1016/j.anihpc.2012.02.004.  Google Scholar

[30]

G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Calc. Var. and PDEs, 42 (2011), 563-577.  doi: 10.1007/s00526-011-0398-7.  Google Scholar

[31]

G. Lu and J. Zhu, Axial symmetry and regularity of solutions to an integral equations in a half-space, Pacific J. Math., 253 (2011), 455-473.  doi: 10.2140/pjm.2011.253.455.  Google Scholar

[32]

G. Lu and J. Zhu, Liouville-type theorems for fully nonlinear elliptic equations and systems in half spaces, Advanced Nonlinear Studies, 13 (2013), 979-1001.  doi: 10.1515/ans-2013-0413.  Google Scholar

[33]

G. Lu and J. Zhu, The maximum principles and symmetry results for viscosity solutions of fully nonlinear equations, J. Differ. Eqs, 258 (2015), 2054-2079.  doi: 10.1016/j.jde.2014.11.022.  Google Scholar

[34]

L. Ma and D. Chen, A Liouville type theorem for an intergral system, Comm. Pure Appl. Anal., 5 (2006), 855-859.  doi: 10.3934/cpaa.2006.5.855.  Google Scholar

[35]

W. Reichel and T. Weth, A prior bounds and a liouville theorem on a half-space for higherorder elliptic Dirichlet problems, Math. Z., 261 (2009), 805-827.  doi: 10.1007/s00209-008-0352-3.  Google Scholar

[36]

J. XuH. Wu and Z. Tan, Radial symmetry and asymptotic behaviors of positive solutions for certain nonlinear integral equations, J. Math. Anal. Appl., 427 (2015), 307-319.  doi: 10.1016/j.jmaa.2015.02.043.  Google Scholar

[37]

J. Villavert, Shooting with degree theory: Analysis of some weighted poly-harmonic systems, J. Differ. Eqs., 257 (2014), 1148-1167.  doi: 10.1016/j.jde.2014.05.003.  Google Scholar

[38]

R. ZhuoF. Li and B. Lv, Liouville type theorems for Schrodinger system with Navier boundary conditions in a half space, Comm. Pure Appl. Anal., 3 (2014), 977-990.  doi: 10.3934/cpaa.2014.13.977.  Google Scholar

[39]

Z. Zhang, Nonexistence of positive solutions of an integral system with weights, Adv. Difference Equ. , 61 (2011), 10 pp. doi: 10.1186/1687-1847-2011-61.  Google Scholar

[40]

Y. Zhao, Regularity and symmetry for solutions to a system of weighted integral equations, J. Math. Anal. Appl., 391 (2012), 209-222.  doi: 10.1016/j.jmaa.2012.02.016.  Google Scholar

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